Unit 3 How Much Can One Sample Tell Us?

The previous unit established that the sample mean is a good estimator of the population mean. Across repeated samples it is centred on the truth, and larger samples produce more precise estimates.

There is a practical problem with how we established it.

Researchers do not observe repeated samples. A survey is conducted once, producing exactly one sample and one estimate. The sampling distribution that justified the sample mean is invisible from inside a single dataset.

So the question of this unit is the one in its title.

How much can a single sample tell us about an unknown population?

This unit develops the tools for answering it. We begin by separating an estimate from the uncertainty surrounding it — a distinction the previous unit made informally and this one makes precise. We then show how the standard error measures the precision of an estimate, and how a confidence interval turns that precision into a range of plausible values for the population parameter.

By the end of this unit you should be able to say not only what a sample suggests about a population, but how confident that suggestion deserves to be — and why an estimate reported without a measure of its uncertainty is an incomplete answer.

The unit stops short of one thing. Evaluating a specific claim about a population — testing whether the average really is ₹20,000 — is the subject of Unit 4. What we build here turns out to be most of the machinery that unit needs.